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Calculation of the Debye temperatures of crystals
]ayarama Reddy, P. 1963
Physica 29 63-66
CALCULATION OF THE DEBYE TEMPERATURES OF CRYSTALS by P. JAYARAMA REDDY Physics Department, Sri Venkateswara University, Tirupati, India.
Synopsis A method of calculating the Debye characteristic temperature of polycrystalline aggregates from a knowledge of the velocities of longitudinal and torsional low frequency waves has been discussed here. The measured velocities of these longitudinal and torsional waves have been taken for the velocities of propagation of plane elastic waves averaged over all directions in the crystal. The calculated values of e are compared with the experimental values.
The Debye characteristic temperature, e is a very important thermal property of the solid. The importance of the parameter has led to the development of methods by which the Debye temperature could be calculated from a knowledge of the elastic properties of the solid. Debye's theory of specific heats shows that the Debye temperature is given by
e=
(h/k)(9N/4nVI)!
(1)
where h is Planck's constant, k is Boltzmann's constant, N is the number of atoms in volume V and I is given by 4,.,
1=12:. (l/vr)(dQ/4n) o
(2)
i
where Vi stands for the velocities of propagation of low frequency waves determined by Christoffel's equation. The problem therefore really reduces to that of averaging the inverse cube of each of the elastic velocities over all directions. For the purpose of averaging, two methods are available: (i) the numerical method using de Launay's tables 1) , which is applied to some cubic crystals, and which involves very tedious calculations, and (ii) the series expansion method, where there are two approaches available, a) Hopf and Lechner's method 2) as modified by Quimby and St u t t on s) and b) Houston's method as developed by Betts and otherss). Results of the later method have recently been used by] oshi and Ml t r a ') to compute the Debye temperatures of a number of solids. These computations are based -
63-
64
P. JAYARAMA REDDY
on the elastic properties of the single crystals of the substance and certain parameters defined by spherical and cubic harmonics. When such computations are made it is found that the calculated values of g differ from values determined from specific heat data by as much as 5 to 20 percent. Betts, Bhatia and Ho r t on-) attribute the discrepancy to two causes; (i) that the specific heat being of an older date may not correspond to the T3-region and (ii) elastic constants measured are not close to OOK.
In view of the difficulties mentioned above, the position has been reviewed from the following points of view. Equation (1) can be written in the form
e=
hVm/k
(3)
where
Vm = vm(3NJ4nV)*
(4)
and 3/v~ =
I: 1/v'tn,
(5)
i
being the three velocities of propagation of plane elastic waves averaged over all directions in the crystal. In a polycrysta1line material, the velocity of the longitudinal wave is given by (6) V~l = 1/p'533
Vrn,
where p is the density and 5 33 is the average longitudinal compliance calculated on the basis of Boas' method 6). Also (7) The average value of the torsion constant is also evaluated on the same basis. Values of umi and Vmt calculated on the above basis are used in equations (5), (4) and (3) to complete the evaluation of e. In recent years polycrystalline aggregates of a number of important substances have been studied and velocities of propagation of longitudinal and torsional waves are directly measured. Where such experimental data are available they are directly used in equations (5), (4) and (3) to obtain the characteristic Debye temperature. Values of e, calculated on this basis from the data at 300 0 R are given in tables I and II and compared against the experimental values. References to relevant literature are shown in the table itself. It is seen from the above two tables, that the agreement between calculated and experimental values is very good. The average difference between experimental and calculated values lies between 0 and 15 percent, barring the very different values in the cases of magnesium, molybdenum, cobalt and silver chloride.
65
THE DEBYE TEMPERATURES OF CRYSTALS TABLE I Polycrystalline data Substance
I
Calculated oK
266 197 167 222 189 253 480 231 133 435 555 544 452 580 91 391 484 134 94 295 230 167 269 124
Silver Tin Cadmium Zinc Gold Platinum Nickel Antimony Bismuth Magnesium Molybdenum Cobalt Tungsten Iron Lead Copper Aluminium Sodium Potassium NaCI KCl KBr NH4Cl
1<.1
I
Experimental 0) oK
Reference
I
229 195 168 0 ) 235 180 b) 233 456 b) 204117 342 420 b) 445 0 ) 379 b) 467 0 ) 88 0 ) 343 418 0 ) 160 99 0 ) 281 230 0 ) 177 270 115-200
Author
" "
" " " " "
" " " "
"
" Bergmann 7) " Hc a r "m o n 8)
" Subrahmanyam 9) " " " Bhagavantam Seshagiri Rao
0) see ref. 12;
0) see ref. 11 j
Single crystal data
Ba(NO.h Pb(NO.h Sr(NOsh AgCl AgBr TIBr NaBr NaCIOs MgO LiP Diamond Galena Pyrites Zinc blende Beryl
I
Calculated
OK 165 122 175 136 128 109 202 236 827 676 2117 220 583 288 195
10)
0) see ref. 13 if not otherwise stated.
TABLE II
Substance
and
Experimental
I
OK
183 144 -
-
750-890 607-750 1860 194 645 300
-
I
Reference Hearman 8)
" " " " " "
"
. "
"
" " "
"
66
THE DEBYE TEMPERATURES OF CRYSTALS
The approach is purely from the experimental side and obviates all the difficulties associated with the processes of averaging from Christoffels' equations. In the cases where the differences are above 10 percent, it may be worthwhile redetermining the heat capacities. I sincerely thank Dr. J. Bhimasenachar for the many useful discussions and encouragement during the course of this work. My thanks are also due to the authorities of Sri Venkateswara University, for the facilities given. Received 26-7-62
REFERENCES 1) 2) 3) 4) 5) 6) 7) B) 9) 10) 11) 12) 13)
De La u n a y, ]., Solid State Physics (Academic Press, New York 1956). Hopf, L. and Lechner, G., Verh. Dtsch, phys. Ges., 16 (1914) 643. Quimby, S. L. and Sutton, P. M., Phys, Rev. 91 (1953) l1'22. Betts, D. D., Bhatia, A. B. and Horton, G. K., Phys. Rev. 104 (1956) 43. Betts, D. D., Bhatia, A. B. and Wyman, M., Phys, Rev. 104 (1956) 37. ] os h i, S. I{. and Mitra, S. S., Proc. phys, Soc. 76 (1960) 295. Boas, W., Introduction to Physics of metals and alloys (John Wiley, New York 1947). Bergmann, L., Der Ultraschall, (1954). HearmoIl, R. F. S., Rev. mod. Phys. 111 (1946) 409. Subrahmanyarn, S. V., Acustica 12 (1962) 37. Bhagavantam, S. and Ses h a g i r i Rae, T., Proc, Ind. Acad. Sci. ariA (1952) 129. Kittel, C., Introduction to Solid State Physics (John Wiley, New York, 1956). Roberts,]. K., Heat and Thermodynamics (Blackies, London, 1951). American Institute of Physics Handbook (Mc-Graw Hill, New York, 1957).
Physica 29 63-66
CALCULATION OF THE DEBYE TEMPERATURES OF CRYSTALS by P. JAYARAMA REDDY Physics Department, Sri Venkateswara University, Tirupati, India.
Synopsis A method of calculating the Debye characteristic temperature of polycrystalline aggregates from a knowledge of the velocities of longitudinal and torsional low frequency waves has been discussed here. The measured velocities of these longitudinal and torsional waves have been taken for the velocities of propagation of plane elastic waves averaged over all directions in the crystal. The calculated values of e are compared with the experimental values.
The Debye characteristic temperature, e is a very important thermal property of the solid. The importance of the parameter has led to the development of methods by which the Debye temperature could be calculated from a knowledge of the elastic properties of the solid. Debye's theory of specific heats shows that the Debye temperature is given by
e=
(h/k)(9N/4nVI)!
(1)
where h is Planck's constant, k is Boltzmann's constant, N is the number of atoms in volume V and I is given by 4,.,
1=12:. (l/vr)(dQ/4n) o
(2)
i
where Vi stands for the velocities of propagation of low frequency waves determined by Christoffel's equation. The problem therefore really reduces to that of averaging the inverse cube of each of the elastic velocities over all directions. For the purpose of averaging, two methods are available: (i) the numerical method using de Launay's tables 1) , which is applied to some cubic crystals, and which involves very tedious calculations, and (ii) the series expansion method, where there are two approaches available, a) Hopf and Lechner's method 2) as modified by Quimby and St u t t on s) and b) Houston's method as developed by Betts and otherss). Results of the later method have recently been used by] oshi and Ml t r a ') to compute the Debye temperatures of a number of solids. These computations are based -
63-
64
P. JAYARAMA REDDY
on the elastic properties of the single crystals of the substance and certain parameters defined by spherical and cubic harmonics. When such computations are made it is found that the calculated values of g differ from values determined from specific heat data by as much as 5 to 20 percent. Betts, Bhatia and Ho r t on-) attribute the discrepancy to two causes; (i) that the specific heat being of an older date may not correspond to the T3-region and (ii) elastic constants measured are not close to OOK.
In view of the difficulties mentioned above, the position has been reviewed from the following points of view. Equation (1) can be written in the form
e=
hVm/k
(3)
where
Vm = vm(3NJ4nV)*
(4)
and 3/v~ =
I: 1/v'tn,
(5)
i
being the three velocities of propagation of plane elastic waves averaged over all directions in the crystal. In a polycrysta1line material, the velocity of the longitudinal wave is given by (6) V~l = 1/p'533
Vrn,
where p is the density and 5 33 is the average longitudinal compliance calculated on the basis of Boas' method 6). Also (7) The average value of the torsion constant is also evaluated on the same basis. Values of umi and Vmt calculated on the above basis are used in equations (5), (4) and (3) to complete the evaluation of e. In recent years polycrystalline aggregates of a number of important substances have been studied and velocities of propagation of longitudinal and torsional waves are directly measured. Where such experimental data are available they are directly used in equations (5), (4) and (3) to obtain the characteristic Debye temperature. Values of e, calculated on this basis from the data at 300 0 R are given in tables I and II and compared against the experimental values. References to relevant literature are shown in the table itself. It is seen from the above two tables, that the agreement between calculated and experimental values is very good. The average difference between experimental and calculated values lies between 0 and 15 percent, barring the very different values in the cases of magnesium, molybdenum, cobalt and silver chloride.
65
THE DEBYE TEMPERATURES OF CRYSTALS TABLE I Polycrystalline data Substance
I
Calculated oK
266 197 167 222 189 253 480 231 133 435 555 544 452 580 91 391 484 134 94 295 230 167 269 124
Silver Tin Cadmium Zinc Gold Platinum Nickel Antimony Bismuth Magnesium Molybdenum Cobalt Tungsten Iron Lead Copper Aluminium Sodium Potassium NaCI KCl KBr NH4Cl
1<.1
I
Experimental 0) oK
Reference
I
229 195 168 0 ) 235 180 b) 233 456 b) 204117 342 420 b) 445 0 ) 379 b) 467 0 ) 88 0 ) 343 418 0 ) 160 99 0 ) 281 230 0 ) 177 270 115-200
Author
" "
" " " " "
" " " "
"
" Bergmann 7) " Hc a r "m o n 8)
" Subrahmanyam 9) " " " Bhagavantam Seshagiri Rao
0) see ref. 12;
0) see ref. 11 j
Single crystal data
Ba(NO.h Pb(NO.h Sr(NOsh AgCl AgBr TIBr NaBr NaCIOs MgO LiP Diamond Galena Pyrites Zinc blende Beryl
I
Calculated
OK 165 122 175 136 128 109 202 236 827 676 2117 220 583 288 195
10)
0) see ref. 13 if not otherwise stated.
TABLE II
Substance
and
Experimental
I
OK
183 144 -
-
750-890 607-750 1860 194 645 300
-
I
Reference Hearman 8)
" " " " " "
"
. "
"
" " "
"
66
THE DEBYE TEMPERATURES OF CRYSTALS
The approach is purely from the experimental side and obviates all the difficulties associated with the processes of averaging from Christoffels' equations. In the cases where the differences are above 10 percent, it may be worthwhile redetermining the heat capacities. I sincerely thank Dr. J. Bhimasenachar for the many useful discussions and encouragement during the course of this work. My thanks are also due to the authorities of Sri Venkateswara University, for the facilities given. Received 26-7-62
REFERENCES 1) 2) 3) 4) 5) 6) 7) B) 9) 10) 11) 12) 13)
De La u n a y, ]., Solid State Physics (Academic Press, New York 1956). Hopf, L. and Lechner, G., Verh. Dtsch, phys. Ges., 16 (1914) 643. Quimby, S. L. and Sutton, P. M., Phys, Rev. 91 (1953) l1'22. Betts, D. D., Bhatia, A. B. and Horton, G. K., Phys. Rev. 104 (1956) 43. Betts, D. D., Bhatia, A. B. and Wyman, M., Phys, Rev. 104 (1956) 37. ] os h i, S. I{. and Mitra, S. S., Proc. phys, Soc. 76 (1960) 295. Boas, W., Introduction to Physics of metals and alloys (John Wiley, New York 1947). Bergmann, L., Der Ultraschall, (1954). HearmoIl, R. F. S., Rev. mod. Phys. 111 (1946) 409. Subrahmanyarn, S. V., Acustica 12 (1962) 37. Bhagavantam, S. and Ses h a g i r i Rae, T., Proc, Ind. Acad. Sci. ariA (1952) 129. Kittel, C., Introduction to Solid State Physics (John Wiley, New York, 1956). Roberts,]. K., Heat and Thermodynamics (Blackies, London, 1951). American Institute of Physics Handbook (Mc-Graw Hill, New York, 1957).